Almost six years after it stumbled into existence, that remains a surprisingly hard question to answer. And with hundreds of millions of users, with many of its social media peers lining up to go public, and with more than a billion dollars of private capital invested, you might think the pressure is building on Twitter to better define itself — and, not incidentally, figure out a way to make more money.

But you’d be wrong. For the moment, Twitter CEO Dick Costolo makes it clear that the company known for instantaneous communication is going to take its own sweet time figuring out what it wants to become.

Running Twitter “doesn’t come with a sense of pressure,” Costolo said this week. “It feels like an opportunity I have to seize.”

Costolo joined the company two years ago as chief operating officer, and took the CEO seat in October 2010 when co-founder Ev Williams stepped aside. He inherited a company closely associated with its founders even as it was still growing at a blistering pace.

As Costolo spoke Thursday, he was standing in the future headquarters of the San Francisco company. The company invited members of the media to get a first look at the space as part of a bigger product announcement.

Twitter plans to occupy three floors in an art deco building in one of the city’s seedier neighborhoods. The city enacted a series of tax cuts earlier this year to entice Twitter to make the move, hoping it would revitalize the area.

The space has been stripped down to its concrete bones inside in preparation for its big renovation. Indeed, many of the windows were smashed or missing, giving the space an arctic chill.

And yet, spending millions of dollars to fix up this space is just one of many signs of the company’s confidence in its future. For all its impact, the company still has a relatively small number of employees: about 700. The new space will accommodate several thousand more.

“We’re going to need those people to scale the company,” Costolo said during a press conference.

Twitter continues to experience explosive growth, at times almost in spite of itself. Twitter can still be unreliable, and new users are frequently baffled about what to do when they first sign up.

The company knows that, and at the event Thursday announced a series of changes to its website and mobile-phone apps to make Twitter easier for non-techies to use it and find content they want. Costolo said the company is trying to think about how the next billion users will join the service by stripping away nonessential features.

In an interview after the announcement, Twitter co-founder Jack Dorsey described the new layout as “crisp, clean, simple.”

Making Twitter easier to use, though, is just one of its big challenges. While Costolo has run several startups, Twitter is already by far the largest company he has run and will get even larger as it hires thousands of employees in the coming years.

To that end, Costolo has been thinking about how to preserve Twitter’s culture by trying to define its values clearly and hiring people with “passion and personality.”

However, Costolo made it clear what those core values don’t include: Making money. Perhaps the most frequently asked question about Twitter’s future (“What’s the business model?”) is the one that seems to concern Costolo the least.

To be clear: Costolo understands that the company will have to generate revenue to support the size and impact he wants the company to have. But despite skepticism about the company’s financial future, he refuses to make that his focus. For now, he’s happy with the measured pace of early advertising products such as letting companies pay to promote tweets and accounts.

“We should think of revenue like oxygen,” Costolo said. “It’s necessary for life, but it’s not the purpose of life. If we do the right things, the businesses are going to follow it.”

Costolo said Twitter has enough money in the bank to sustain it for the foreseeable future. There is no sense of IPO-envy as companies like Zynga, LinkedIn, Groupon and Facebook graduate to the public markets.

“We can stay private and grow the business the way we want, as long as we want,” Costolo said. “We never think about or talk about when we want to go public.”

Twitter has already made a tremendous impact on culture and politics. But Costolo may be aiming for something even loftier: to transcend the profit-driven motivation that propels most companies.

As he puts it: “Twitter has the opportunity to be one of the great companies in the world.”

Many talk about it, but few truly mean it. If Costolo fulfills that promise, that may be the biggest impact yet that he and Twitter can make.